Construction of Statistically Similar Representative Volume Elements – Comparative Study Regarding Different Statistical Descriptors

Abstract

Advanced high strength steels, such as dual-phase steel (DP steel), provide advantages for engineering applications compared to conventional high strength steel. The main constituents of DP steel on the microscopic level are martensitic inclusions embedded in a ferritic matrix. A way to include these heterogeneities on the microscale into the modeling of the material is the FE2- method. Herein, in every integration point of a macroscopic finite element problem a microscopic boundary value problem is attached, which consists of a representative volume element (RVE) often defined as a segment of a real microstructure. From this representation, high computational costs arise due to the complexity of the discretization which can be circumvented by the use of a Statistically Similar RVE (SSRVE), which is governed by similar statistical features as the real target microstructure but shows a lower complexity. For the construction of such SSRVEs, an optimization problem is constructed which consists of a least-square functional taking into account the differences of statistical measures evaluated for the real microstructure and the SSRVE. This functional is minimized to identify the SSRVE for which the similarity in a statistical sense is optimal. The choice of the statistical measures considered in the least-square functional however play an important role. We focus on the construction of SSRVEs based on the volume fraction, lineal-path function and spectral density and check the performance in virtual tests. Here the response of the individual SSRVEs is compared with the real target microstructure. Further higher order measures, some specific Minkowski functionals, are investigated regarding their applicability and efficiency in the optimization process.